Question for step mommas

For those of you that have to deal with a high conflict birth mom. How do you deal with her insecurities being taken out on your SO. I recently found out that she is now all the sudden saying that my SO and i are making it seem like we are step daughter's parents and not her. Step daughter knows im not her mom and calls me by my name always has since she learned to talk. Her mom has figured out that step daughter listens to me and my SO when she is being bad. But step daughter will not listen to her mom or step dad when she is bad at their home. Loke step daughter is a complete different person with us than with them. She will flat out refuse and tell them no when they ask her to do something or even tell herand her mom has punished her and spanked her pleanty of times. She doesnt do that with us, when we ask her and she doesnt do it she gets a warning usually by the warning she does what she is told to do because she will not test my SO and doesnt test me because my SO does the punishments. She knows something will happen if she does (usually he says he will throw away her toys, time out, and he does say he will spank her but never has had to and he really doesnt believe in spanking). now all these accusations are coming up that are not true. It doesnt bug me but because my SO is getting an ear full everytime she complains im concerned it may take a toll on him. He is not one to talk about that stuff or have me worry about it so he just keeps it to himself. Ive told him he can talk to me of he needs to blow off some steam and he does but very rarely and i can tell when he is to the point of being fed up with her because he tends to get snappy with me but he just doesnt share. He also defends me alot to his ex because with her its either im not doing enough or im doing too much. I do alot with step daughter because she is literally attached to my hip when she is with me and my SO. I dont want my SO to have to fight a battle that isnt really his to fight because her issue is with me and hiw close i am to step daughter. I honestly would be happy if he just ignored it rather than taking on that burden or tell his ex of she has a problem with me she can talk to me about it.